The Datasets

• Movement: contains the location and datetime of the 41 collared deer in the period Feb 2020 - Feb 2023 in the Bavarian Forest National Park. Movement is tracked at hourly intervals.

• Hunting Events: contains location and date of hunting events in the Bavarian Forest National Park - in total 1270 events, 890 of them with full timestamp

• FCM Stress: contains information of 809 faecal samples, including:
  

  • the FCM level [ng/g]
    
  • the location of the sample
    
  • the time of sampling
    
  • the DNA-matched collared deer
    
  • the time when the deer was at the location

• Reproduction Success: observations of 16 collared deer on:
  

  • if they were pregnant in one year
    
  • if they were accompanied by a calf in one year

The Data-Generating Process

• A deer roams freely in the forest. Its movement is tracked by GPS collar.
• A hunting event happens.
• After some time, the deer defecates. This is a defecation event.
• After some time, researchers go to the defecation location and collect an FCM sample.

Note: The defecation location is not the deer’s location at the time of the stress event.

Relevant Hunting Events

We introduce 4 Parameters:

• Gut Retention Time (GRT) low [hours],
• Gut Retention Time (GRT) high [hours],
• Distance Threshold [km],
• Proximity Criterion (“last”, “nearest”, or “higest score”)

A hunting event is considered relevant to an FCM sample, if

• the time difference (“TimeDiff”) between experiencing stress (hunting) and defecation is in [GRT low, GRT high], and
• the distance between the deer and the hunting event is \(\leq\) distance threshold at the time of hunting.

The Most Relevant Hunting Event

Among the relevant hunting events, the most relevant one is defined by the proximity criterion:

• the closest in time (“last”),
• the closest in space (“nearest”), or
• the one with the “highest score.”

The scoring function is defined as TBD.

Illustration

The Models

For Modelling, we consider the following covariates:

• Time difference [hours]
• Distance [km]
• Sample delay [hours]
• Pregnant
• Defecation day (between 1 and 366)
• Number of other hunting events

The Models

A Generalized Additive Model

i) Gaussian: Identity Link

• \(FCM_i \sim \mathcal{N}(\mu_i, \sigma^2)\)

• Identity Link: \(E(FCM_i) = \mu_i = \eta_i\)

• Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

i) Gaussian: Identity Link

A Generalized Additive Model

ii) Gamma: Log Link

• \(FCM_i \sim \mathcal{Ga}(\nu, \frac{\nu}{\mu_i})\)

• For better Interpretability we use the Log-Link: \(E(FCM_i) = \mu_i = exp(\eta_i)\)

• Linear Predictor: \[ \begin{equation} \begin{gathered} \eta_i = \beta_0 + \beta_1\,Pregnant_i +\\ \beta_2\,Number\,Other\,Hunts_i + f_1(Time\,Diff_i) + \\ f_2(Distance_i) + f_3(Sample\,Delay_i) + f_4(Day\,of\,Year_i) \end{gathered} \end{equation} \]

A Generalized Additive Model

ii) Gamma: Log Link

A Generalized Additive Model

ii) Gamma: Log Link

A Generalized Additive Model

ii) Gamma: Log Link

A Generalized Additive Model

ii) Gamma: Log Link

A Generalized Additive Model

ii) Gamma: Log Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer.

\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{N}\left( \mu_{ij}, \sigma^2 \right) \\ \mu_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + \\ && f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\sim& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gaussian: Identity Link

B Generalized Additive Mixed Model

i) Gamma: Log link

• Log link for interpretability.

• Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer.

\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}) \\ \eta_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + \\ && f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\sim& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]

B Generalized Additive Mixed Model

ii) Gamma: Log Link

B Generalized Additive Mixed Model

ii) Gamma: Log Link

B Generalized Additive Mixed Model

ii) Gamma: Log Link

B Generalized Additive Mixed Model

ii) Gamma: Log Link

B Generalized Additive Mixed Model

ii) Gamma: Log Link